Trying to make a circle in Minecraft using coordinates and Sin & Cos - math

Question

I am trying to write a Minecraft Datapack, which will plot a full armorstand circle around whatever runs the particular command. I am using a 3rd party mathematics datapack to use Sin and Cos. However, when running the command, the resulting plot was... not good. As you can see here: 1. Broken Circle., rather than have each vertex evenly placed in a circular line, I find a strange mess instead.
I would have thought loosing precision in Cos and Sin would simply make the circle more angular, I didn't expect it to spiral. What confuses me, is that +z (the red square) and -x (the purple one) are all alone. You can see on the blue ring (Which was made with a smaller radius) the gap between them persists.
My main issue is; How did my maths go from making a circle to a shredded mushroom, and is there a way to calculate the vertices with a greater precision?
Going into the project I knew I could simply spin the centre entity, and summon an armorstand x blocks in front using ^5 ^ ^, however I wanted to avoid this, due to my desire to be able to change the radius without needing to edit the datapack. To solve this, I used the Sin and Cos components to plot a new point, using a radius defined with scoreboards.
I first tested this using Scratch, in order to check my maths. You can see my code here: 2. Scratch code.
With an addition of the pen blocks, I was able to produce a perfect circle, which you can see here:
. Scratch visual proof.
With my proof of concept working, I looked online and found a Mathematical Functions datapack by yosho27, since the Cos and Sin functions are not built into the game. However, due to how Minecraft scoreboards are only Integers, Yosho27 multiplied the result of Cos and Sin by 100 to preserve 2 decimal places.
To start with, I am using a central armorstand with the tag center, which is at x: 8.5 z: 8.5. The scoreboards built into yosho's datapack that I am using is math_in for the values I want converted and math_out, which is where the final value is dumped.
Using signs, I keep track of the important values I am working with, as seen here: 4. Sign maths.
As I was writing this, I decided to actually compare both numbers to find this: 5. Image comparison, which shows me that somewhere in this calculation process, the maths has gone wrong. I modified the scratch side to match the minecraft conditions as much as possible, such as x100 and adding 850 to the result. From this result, I can see a disparity between x and z, even though they should be equal. Where Minecraft says 1: x= 864 z= 1487, Scratch says 1: x= 862.21668448: z= 1549.89338664. I assume this means the datapack's Cos and Sin are not accurate enough?
In light of this , I looked in yosho's datapack, I found this: 6. Yosho's code., which I just modified to be *= 10 instead of divide, in the hope of getting more precision. Modifying the rest of my code to match, I couldn't see any improvement in the numbers, although the armorstand vertices were a few pixels off the original circle, although I couldn't find a discernible pattern to this shift.

Answer 1

While this doesn't answer your full question, I'd like to point out two different ways you can solve the original issue at hand, no need to rely on some foreign math library:
^ ^ ^
Use Math, but let the game do it for you.
You can use the fact that the game is doing those rotational conversions for you already when using local coordinates. So, if you (or any entity) go to 0 0 0 and look / rotate in the angle that you want to calculate, then move forward by ^ ^ ^1, the position you're at now is basically <sin> 0 <cos>.
You can now take those numbers with your desired precision using data get and continue using them in whatever way you see fit.
Use recursive functions to move in incremenets
You point out in your question that
Going into the project I knew I could simply spin the centre entity, and summon an armorstand x blocks in front using ^5 ^ ^, however I wanted to avoid this, due to my desire to be able to change the radius without needing to edit the datapack. To solve this, I used the Sin and Cos components to plot a new point, using a radius defined with scoreboards.
So, to go back to that original idea, you could fairly easily (at least easier than trying to calculate the SIN/COS manually) find a solution that works for (almost) arbitrary radii and steps: By making the datapack configurable through e.g. scores, you can set it up to for example move forward by ^^^0.1 blocks for every point in a score, that way you can change that score to 50 to get a distance of ^^^5 and to 15 to get a distance of ^^^1.5.
Similarly you could set the "minimum" rotation between summons to be 0.1 degrees, then repeating said rotation for however many times you desire.
Both of these things can be achieved with recursive functions. Here is a quick example where you can control the rotational angle using the #rot steps score and the distance using the #dist steps score as described above (you might want to limit how often this runs with a score, too, like 360/rotation or whatever if you want to do one full circle). This example technically recurses twice, as I'm not using an entity to store the rotation. If there is an entity, you don't need to call the forward function from the rotate function but can call it from step (at the entity).
step.mcfunction
# copy scores over so we can use them
scoreboard players operation #rot_steps steps = #rot steps
scoreboard players operation #dist_steps steps = #dist steps
execute rotated ~ ~0.1 function foo:rotate
rotate.mcfunction
scoreboard players remove #rot_steps steps 1
execute if score #rot_steps matches ..0 positioned ^ ^ ^.1 run function foo:forward
execute if score #rot_steps matches 1.. rotated ~ ~0.1 run function foo:rotate
forward.mcfunction
scoreboard players remove #dist_steps steps 1
execute if score #dist_steps matches ..0 run summon armor_stand
execute if score #dist_steps matches 1.. positioned ^ ^ ^.1 run function foo:forward